Secondary menu

Search form

Maths in a minute: negative numbers

Submitted by mf344 on January 21, 2011

<

Negative numbers are easy to imagine if you think of the number line as
a giant thermometer which includes sub-zero temperatures. This makes
addition and subtraction easy, as you just move up or down the number
line by the according amount.

But what about those tricky multiplication rules? Why does positive
times negative give negative, and negative times negative give positive?
Here the number line can help us too.

Suppose you're standing at the point 0, facing in the positive direction
of the number line. You take two steps backwards and you do this 4 times.
You end up at the point -8, showing that -2 steps times 4 is -8, ie (-2)x4=-8.

Now suppose you're back at 0, this time facing in the negative direction.
You take 2 steps forwards and you do this 4 times. You also end up at
point -8, showing that 2 steps times -4 is -8, ie 2x(-4)=-8.

Again, go back to 0, looking in the negative direction. Take 2 steps
backwards and do this 4 times. You end up at the point 8. Stepping
backwards gives you a -2. Facing in the negative direction gives you a
-4. Putting all this together gives (-2)x(-4)=8.

I find it easier to visualise a number as consisting of the number's value plus an infinite number of +1, -1 pairs. Then, for me, subtracting negative numbers is just a matter of removing the appropriate number of -1's from the pairs and seeing what's left.
So 5 - -1=(5+1-1)- -1=5+1=6
(-2)x4=subtract 2 lots of +4 from the +1-1 pairs=-8
(-2)x(-4)=subtract 2 lots of -4 from the +1-1 pairs=+8